Gauss s Law. Lecture 4. Chapter 27. Channel 61 (clicker) Physics II

Transcription

1 Lecture 4 Chapter 27 Physics II Gauss s Law Course website: Lecture Capture: Channel 61 (clicker)

2 The idea behind Gauss s law It looks like number of lines passing through a closed surface are related to the amount of charge inside (Gauss s Law) But, first, we need to learn how to count lines

3 The Basic Definition of Flux Imagine holding a rectangular wire loop of area A in front of a fan. The volume of air flowing through the loop each second depends on the angle between the loop and the direction of flow. The flow is maximum through a loop that is perpendicular to the airflow. No air goes through the same loop if it lies parallel to the flow.

4 The area vector Let s define an area vector to be a vector in the direction of, perpendicular to the surface, with a magnitude A equal to the area of the surface. Vector has units of m 2.

5 The Electric Flux is a measure of the number of field lines passing through a surface A Consider 1) a uniform electric field E 2) a flat surface The electric flux through a surface of area A can be defined as the dot-product: θ

6 The Electric Flux (general case) Consider 1) a non-uniform electric field E 2) area is not flat (curved) Divide the surface into many small pieces of area A. a) Each piece is small enough that it is essentially flat b) The field is nearly uniform over each piece c) Thus, the formula from the previous slide can be used The electric flux through each small piece is: The electric flux through the whole surface is the surface integral:

7 The Electric Flux through a closed surface Consider 1) a non-uniform electric field E 2) a closed surface NOTE: For a closed surface, we use the convention that the area vector da is defined to always point toward the outside. The electric flux through a closed surface: closed surface

8 Channel 61 ConcepTest 1 Electric Flux The electric flux through the shaded surface is A) 0 B) 400cos20 N m 2 /C C) 400cos70 N m 2 /C D) 400 N m 2 /C E) Some other value

9 Gauss s Law For any closed surface enclosing total charge Q in, the net electric flux through the surface is: Φ Gaussian surface 1) It works for any closed Gaussian surface 2) Q in is the net charge enclosed by the Gaussian surface (charges outside must not be included) 3) Distribution of Q in doesn't matter 4) A Gaussian surface is an imaginary, mathematical surface q 1 q 4 q 2 q 5 q 3 Gaussian surface Both, Gauss s law and Coulomb s law, help to find electric fields based on distribution of charges.

10 Gauss s Law/Symmetry Gauss s law is always true, but it is not always useful Φ Gaussian surface Evaluation of this surface integral is often difficult. However, when the charge distribution has sufficient symmetry (spherical, cylindrical, planar), evaluation of the integral becomes simple. Spherical symmetry Cylindrical symmetry Planar symmetry

11 Channel 61 ConcepTest 2 Electric Flux Which spherical Gaussian surface has the larger electric flux? A) Surface A B) Surface B C) They have the same flux D) Not enough information to tell Flux depends only on the enclosed charge, not the radius.

12 Channel 61 Example Electric Flux Determine the electric flux through each surface Φ Flux depends only on the enclosed charge, not the radius. S1: Φ=(+Q-3Q)/ε 0 S2: Φ=(+Q+2Q-3Q)/ε 0 S3: Φ=(+2Q-3Q)/ε 0 S4: Φ=0 (no charge inside) S5: Φ=(+2Q)/ε 0

13 Examples

14 E field outside of a charged dielectric sphere A total charge Q is spread uniformly throughout a dielectric sphere of radius R. What is the electric field outside the sphere (r>r)? Notice a remarkable feature of this result: The field outside the sphere is exactly the same as it would have been if all the charge had been concentrated at the center.

15 E field inside of a charged dielectric sphere A total charge Q is spread uniformly throughout a dielectric sphere of radius R. What is the electric field inside the sphere (r<r)? Let s plot it E

16 E field of a plane of charge Find the electric field of an infinite nonconducting plane of charge with surface charge density η (C/m 2 ).

17 What you should read Chapter 27 (Knight) Sections 27.2 Electric flux concept 27.3 Electric flux 27.4 Gauss s law 27.5 Examples

18 Thank you See you on Tuesday